Theorem calemos 2322

Description: "Calemos", one of the syllogisms of Aristotelian logic. All φ is ψ (PaM), no ψ is χ (MeS), and χ exist, therefore some χ is not φ (SoP). (In Aristotelian notation, AEO-4: PaM and MeS therefore SoP.) (Contributed by David A. Wheeler, 28-Aug-2016.) (Revised by David A. Wheeler, 2-Sep-2016.)

Hypotheses

Ref	Expression
calemos.maj	⊢ ∀x(φ → ψ)
calemos.min	⊢ ∀x(ψ → ¬ χ)
calemos.e	⊢ ∃xχ

Assertion

Ref	Expression
calemos	⊢ ∃x(χ ∧ ¬ φ)

Proof of Theorem calemos

Step	Hyp	Ref	Expression
1		calemos.e	. 2 ⊢ ∃xχ
2		calemos.min	. . . . . . 7 ⊢ ∀x(ψ → ¬ χ)
3	2	spi 1753	. . . . . 6 ⊢ (ψ → ¬ χ)
4	3	con2i 112	. . . . 5 ⊢ (χ → ¬ ψ)
5		calemos.maj	. . . . . 6 ⊢ ∀x(φ → ψ)
6	5	spi 1753	. . . . 5 ⊢ (φ → ψ)
7	4, 6	nsyl 113	. . . 4 ⊢ (χ → ¬ φ)
8	7	ancli 534	. . 3 ⊢ (χ → (χ ∧ ¬ φ))
9	8	eximi 1576	. 2 ⊢ (∃xχ → ∃x(χ ∧ ¬ φ))
10	1, 9	ax-mp 5	1 ⊢ ∃x(χ ∧ ¬ φ)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)